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We derive new integral estimates on substatic manifolds with boundary ofhorizon type, naturally arising in General Relativity. In particular, wegeneralize to this setting an identity due to Magnanini-Poggesi leading to theAlexandrov Theorem in the Euclidean space and improve on a Heintze-Karcher typeinequality due to Li-Xia. Our method relies on the introduction of a new vectorfield with nonnegative divergence, generalizing to this setting the P-functiontechnique of Weinberger.

New Integral Estimates in Substatic Riemannian Manifolds and the Alexandrov Theorem